This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A387649 #16 Sep 06 2025 15:49:34
%S A387649 1,0,1,6,1,20,21,42,141,128,505,782,1465,3852,5501,14290,26053,49464,
%T A387649 113313,195446,433585,841732,1629285,3477754,6520445,13583472,
%U A387649 26955209,52921182,109117737,212250492,431217101,861537698,1706182325,3460479528,6839829361
%N A387649 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(2*k+1,2*n-4*k).
%H A387649 Vincenzo Librandi, <a href="/A387649/b387649.txt">Table of n, a(n) for n = 0..1500</a>
%H A387649 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,4,-1,4,-4).
%F A387649 G.f.: (1-x^2+2*x^3)/((1-x^2+2*x^3)^2 - 8*x^3).
%F A387649 a(n) = 2*a(n-2) + 4*a(n-3) - a(n-4) + 4*a(n-5) - 4*a(n-6).
%t A387649 Table[Sum[2^(n-2*k)*Binomial[2*k+1,2*n-4*k],{k,0,Floor[n/2]}],{n,0,40}] (* _Vincenzo Librandi_, Sep 06 2025 *)
%o A387649 (PARI) a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(2*k+1, 2*n-4*k));
%o A387649 (Magma) [&+[2^(n-2*k)* Binomial(2*k+1, 2*n-4*k): k in [0..Floor (n/2)]]: n in [0..40]]; // _Vincenzo Librandi_, Sep 06 2025
%Y A387649 Cf. A387624, A387650.
%Y A387649 Cf. A387647.
%K A387649 nonn,new
%O A387649 0,4
%A A387649 _Seiichi Manyama_, Sep 05 2025